Team:TU Delft/Model/Q3

iGEM TU Delft

Modeling

Question 3:

At what fluorophore concentration and cavity size do we reach the laser threshold, when we take the kinetics and dynamics of photons inside the biolaser into account?

Introduction

In order to get lasing cells we encapsulate E. coli in polysilicate or tin dioxide and we use fluorophores as a gain medium. In Q1 we already determined the minimal size in order to fit one wavelength of light inside the cells when the light resonates in whispering gallery modes. In this section we determine the lasing threshold. The laser threshold is the point at which lasing starts occurring. Therefore we take losses due to the mirrors and absorption and the gain due to stimulated emission into account. We will determine here what the minimal size and the minimal concentration of fluorophores is to get lasing.

Laser Kinetics and Dynamics

To achieve lasing, the gain of photons in the system through stimulated emission has to be larger than the loss through absorption and escape. The rate of stimulated emission therefore has to be higher than a critical value, which we call the lasing threshold. In this section we determine this threshold and whether it will be reached in our system. To do so we describe the dynamics of the number of photons with the emission wavelength (N1), the number of GFP molecules in the excited state (GFP1) and the number of GFP molecules in the ground state (GFP0).

Lasers make use of the principle of spontaneous and stimulated emission. When light is absorbed by an atom (or molecule), the energy of that atom increases with the energy of the photon that is adsorbed, to reach a higher energy level (S1) (figure 1A) (Lakowicz, 2007). The atom will not stay in this higher energy level, but through a series of rapid non-radiative transitions its energy is lowered to a metastable state (S1,0). From the metastable state the atom will be able to release its energy in the form of an emitted photon. For spontaneous emission the emission spectrum is often broad and overlaps with the absorption spectrum. The release of energy does not necessarily go by emission of a photon as also non-radiative relaxation can take place. In that case the energy is released by heat.

Figure 1: Energy diagram spontaneous emission (A) and stimulated emission (B). (Super-Resolution Imaging Center)

Next to spontaneous emission, the metastable state of the atom can also be relaxed through stimulated emission, which is the basic principle that allows lasers to work (Hecht, 2002, Svelto et al,2010). In stimulated emission a photon hits the metastable particle and forces it to release its energy as another photon (figure 1B). This new photon has the same characteristics as the incident photon, meaning that they are in phase, with the same polarization, same direction, and same wavelength (Hecht, 2002, Svelto et al, 2010). Stimulated emission can only take place when enough atoms are in the excited state, otherwise normal absorption is much more likely to occur. As a laser depends on stimulated emission, the photons have to be trapped inside a cavity so that they will pass the excited molecules many times and increase the chance of stimulated emission.

In a conventional laser system the gain medium (the fluorophores in our case) is placed between 2 mirrors. Between these mirrors photons can oscillate to create a burst of photons emitted by stimulated emission. However the mirrors are never perfect and therefore some photons are lost from the system. Furthermore the medium between the mirrors (which in our case is the cytoplasm) does absorb photons, so we also should take absorption losses into account. We can describe the dynamics of the fluorophores in excited state (GFP1) and the photons N1 in our system by equations (1,2).

$$ \frac{\partial GFP_1}{\partial t} = Nonradiative Relaxation + (Spontaneous +Stimulated) Emission$$ $$ \frac{\partial N_1}{\partial t} = (Spontaneous +Stimulated) Emission - Mirror losses- Absorption losses$$

Because the total GFP concentration (GFPt) is fixed (we assume that the production and degradation of fluorescent proteins reaches steady state within a short period, see Q2), the dynamics of the ground state GFP molecules follows directly from equation (1,3).

$$ GFP_T=GFP_0+GFP_1$$

In lasing experiments done by other groups (Humar et al, 2015, Gather et al, 2011), with fluorophores as a gain medium, a pulsing laser is used to excite the fluorophores in the system. A pulsing laser is used to create a population inversion In the gain medium of a laser, the atoms can be in either the ground state (lower energy – the equilibrium state) or the excited state (higher energy). Population inversion is the transition of most of the atoms in the system to the opposite state, by the insertion of external energy. The term is often used to indicate almost all atoms have been put in the excited state. while minimizing photo-bleaching. In our model we will look at the photon dynamics directly after a pulse, and therefore population inversion has taken place. We will assume that initially all the GFP molecules are in the excited state. Furthermore we assume that initially there are no photons at emission wavelength (N1) in the system. Therefore the initial conditions are given by equation (4,5).

$$GFP_1(0)=GFP_t$$ $$ N_1(0)=0$$

The non-radiative emission and spontaneous emission in equation (1) are both dependent on the relaxation time of GFP. The spontaneous relaxation rate of GFP1 can be given by equation (6) where ksp is the spontaneous emission rate and \(\tau_{sp}\) the spontaneous emission time (Svelto et al, 2010).

$$k_{sp} = \frac{1}{\tau_{sp}}$$

Because not all the spontaneous relaxation transitions result in emission of a photon, we should take the efficiency of the fluorophores into account. The efficiency of fluorophores can be described by the quantum yield (QY), the number of photons emitted per photon absorbed by a GFP molecule (Lakowicz, 2007). Therefore the spontaneous emission term in equation (1,2) becomes \(QY\cdot k_{sp}\).

The stimulated emission rate kstim (equation 7) is proportional to the stimulated emission cross section (\(\sigma_e\)) of the fluorophore and the volume of the system (Svelto et al, 2010). Here V is the volume of the optical cavity.

$$ k_{stim} = \frac{\sigma_ec}{V}$$

The stimulated emission stimulated emission cross-section The stimulated emission cross-section is a characteristic area of the fluorescent molecule at which upon incidence of a photon, the molecule in the excited state can release a second photon. is comparable to the absorption cross section (\(\sigma_a\)) (Min et al, 2009, Svelto et al., 2010 ), therefore we take \(\sigma_e=\sigma_a\), as literature values of the absorption cross section can be easily found.

The losses due to absorption can be approximated by the law of Lambert-Beer. The time dependent version of Lambert-Beer follows from the differential Lambert-Beer law (Parnis et al. , 2013) and the speed of the photons, c (equation 8). In equation 8, x is the distance traveled, I0 is the intensity of the light, \(\alpha\) the absorption coefficient and N1 the number of photons with the emission wavelength.

$$ \left.\begin{matrix} \frac{\partial I_0}{\partial x} = -\alpha I_0 \\ \frac{\partial x}{\partial t} = c\\ I_0\propto N_1 \end{matrix}\right\} \frac{\partial N_{1,absorption}}{\partial t} = -\alpha \cdot c \cdot N_1 = K_{abs} \cdot N_1 $$

The absorption constant \(\alpha\) is determined experimentally by measuring the absorption of light by our sample at emission wavelength. The total absorption A is given by \(A=\alpha l=\log_{10}\frac{I_0}{I}\). The well we used has a path length of 10.2 mm. Thus resulting in a value for \(\alpha\) of \(15.2 m^{-1}\).

The losses due to the mirrors depend on the number of times a photon hits a mirror and the reflectivity of each mirror (Wilmsen et al, 2001, Svelto et al, 2010). The mirror loss rate (kmirror) is given by equation 9 where c is the speed of light and l the length of the optical cavity; its value is negative because the reflectivity of the mirrors R is always smaller than 1: \(R_1, R_2 <1\). We will determine the reflectivity below.

$$ K_{mirror} = \frac{c}{2l}\ln(R_1R_2)$$

Combining all contributions, the dynamics of the system can be described by the following set of differential equations:

$$\frac{dGFP_1}{dt} = (k_{sp} + k_{stim}\cdot N_1)GFP_1 = \Big(\frac{1}{\tau_sp} +\frac{\sigma_a c}{V} N_1\Big)GFP_1$$ $$\begin{align} \frac{dN_1}{dt} = (QY\cdot k_{sp}+&k_{stim}\cdot N_1)GFP_1+(k_{mirror}+k_{abs})N_1 \nonumber \\ &= \Big(\frac{QY}{\tau_{sp}}+\frac{\sigma_a c}{V}N_1\Big)GFP_1+ \Big(\frac{c}{2l}\ln(R_1R_2)-\alpha\cdot c\Big)N_1 \end{align}$$

Reflectivity of the mirrors

In literature, GFP has been used as a gain medium when placed between two mirrors (Gather et al, 2011), which is in principle a Fabry-Pérot cavity, one of the most used type of laser cavities(Svelto et al, 2010). In this case the model given by equations 10 and 11 can directly be applied to the system.

However, in our project we do not have two mirrors at opposite sides of the cavity, instead we have a spherical cavity. Because the reflectivity of the mirror depends on the angle of incidence, the shape of the cavity has to be taken into account.

In the case of a single cell laser, the layer of polysilicate/tin dioxide is the analog for the mirrors in the Fabry-Pérot system. Although we do not know what the thickness of this layer is, we do know that the thickness of the layer is much smaller than the wavelength \(d\ll \lambda\) and therefore we will model the interface as a thin film (Mcintyre et al, 1971). The calculation of the reflection coefficient of a thin film is a known problem in optics, and can be found in standard textbooks such as (Hecht, 2001).

The relevant equations can be derived from the boundary conditions at the interface of the optical medium. In the simplest case this medium is fully described by its refraction index n. The boundary conditions follow directly from the Maxwell equations, and ensure that the electric and magnetic fields behave appropriately at the interface (Hecht, 2001) .

The fraction of reflected power R is given by

$$R=|r|^2,$$

where r is the amplitude reflection coefficient accounting for the phase shift and can be a complex value. The parameter r can be determined from the electric (\(\vec{E}\)) and the magnetic (\(\vec{H}\)) field and the characteristic matrix, M, relating the fields at the boundaries. For a single thin film we get the following matrix equation.

$$\begin{bmatrix} E_I\\ H_I \end{bmatrix} = M\begin{bmatrix} E_{II}\\ H_{II}\end{bmatrix}$$
Figure 2: Fields at the boundary of a thin film (Hecht, 2001)).

Here EI and HI are the electric and magnetic fields before the thin film and EII and HII are the electric and magnetic field after the thin film as in figure 2. The matrix M is given by:

$$M=\begin{bmatrix} \cos k_0 h & (i\sin k_0h/\Upsilon_1)\\ \Upsilon_1 i\sin k_0h & \cos k_0h \end{bmatrix}$$

Here the k0 is the wave number, \(k_0=\frac{1}{\lambda}\), and h is the shift in phase, \(h=\frac{2n_1d\cos(\theta_{iII})}{2}\) where \(n_1\) is the refractive index of the film. We have to take two kinds of polarization into account: S-polarized light, where the electric field is normal to the plane of incidence, and P-polarized light, where the electric field is parallel to the plane of incidence. The \(\Upsilon_I\) accounts for these different polarizations and for S- and P-polarized light, which is given by equation 15 and 16, respectively.

$$\Upsilon_1 = \sqrt{\frac{\epsilon_0}{\mu_0}}n_1 cos \theta_{iII}$$ $$\Upsilon_1 = \sqrt{\frac{\epsilon_0}{\mu_0}}n_1 /cos \theta_{iII}.$$

We set \(\Upsilon_0 \) and \(\Upsilon_s\) as equation 17 and 18.

$$ \Upsilon_0 = \sqrt{\frac{\epsilon_0}{\mu_0}}n_0\cos\theta_{tI}$$ $$ \Upsilon_s = \sqrt{\frac{\epsilon_0}{\mu_0}}n_s\cos\theta_{tII}$$

Using equation 17 and 18 together with the boundary conditions \(E_I = E_{iI}+E_{rI} = E_{tI} +E_{rII}’\), \(H_I = \sqrt{\frac{\epsilon_0}{\mu_0}} (E_{tI} - E_{rII}’)n_1 \cos\theta_{iII}\) and \(H_{II} = \sqrt{\frac{\epsilon_0}{\mu_0}} E_{tII}n_s\cos\theta_{tII}\), we can rewrite equation 13:

$$\begin{bmatrix} (E_{iI}+E_{rI}) \\ (E_{iI}-E_{rI})\Upsilon0 \end{bmatrix} = M \begin{bmatrix} E_{tII}\\ E_{tII}\Upsilon_s \end{bmatrix}$$

The amplitude coefficient of reflection, r, can then be determined as \(r=\frac{E_{rI}}{E_{iI}}\) from equation 14 and equation 19 and results in equation 20. Equation 20 is determined for both polarizations by using \(\Upsilon_1\) from equation 15 and equation 16 to determine the characteristic matrix.

$$r=\frac{\Upsilon_0m_{11}+\Upsilon_0\Upsilon_sm_{12}-m_{21}-\Upsilon_sm_{22}}{\Upsilon_0m_{11}+\Upsilon_0\Upsilon_sm_{12}+m_{21}+\Upsilon_sm_{22}},$$

The reflectivity of unpolarized light can then be determined as the average reflectivity of both polarizations.

$$R = \frac{1}{2}(R_s+R_p)$$

This reflectivity is now a function of the incidence angle as can be seen in figure 3. In this model we will assume that the angle of incidence is uniformly spread and we will take the reflectivity as the average reflectivity over all the angles. We then find the reflectivity and other parameters used in this model in table 1.

Figure 3: Reflectivity of reflective layer of polysilicate and tin dioxide as a function of angle of incidence.
Table 1: Parameters used in model
Parameter Value Description Source
\(c\) 299792458 m/s Speed of light ( Zangwill, 2013 )
\(\alpha\) \(15.2\) m\(^{-1}\) Absorption coefficient of the cell Computed from measured Absorption (A = 0.1554 \pm 0.0014\
\(\lambda\) 511 nm Emission wavelength GFP (Cormack et al.,1996)
\(\sigma_a=\sigma_e\) \(2.03\cdot 10^{-16} cm^2\) Absorption/ stimulated emission cross section EGFP ( Peterman et al., 1994)
\(QY\) 0.6 Quantum Yield EGFP ( Patterson et al., 2001)
\(\tau_{sp}\) 2.71 ns Spontaneous fluorescence lifetime ( Berezin et al., 2010)
\(\epsilon_0\) \(8.854 187 817\cdot 10^{-12} F m^{-1}\) Vacuum permittivity ( Zangwill, 2013)
\(\mu_0\) \(4\pi \cdot 10^{-7} N/A^2\) Vacuum permeability ( Zangwill, 2013 )
\(n_0\) 1.37 Refractive index cytosol (Liang et al, 2007 )
\(n\)1tin dioxide 2 Refractive index Tin Dioxide ( Patnaik, 2003)
\(n\)1polysilicate 1.47 Refractive index polysilicate ( Polini et al. , 2012)
\(n_s\) 1.33 Refractive index PBS ( Schoch et al. , 2012 )
\(R\)polysilicate 0.2711 Reflectivity thin layer of polysilicate (d=50 nm) From model
\(R\)tin dioxide 0.4607 Reflectivity thin layer of Tin dioxide (d=50 nm) From model

Results

To get stimulated emission there should be a high enough concentration of fluorophores and the cavity should be large enough. In this section we will first verify our model by comparing it with experiments found in literature. Afterwards we will model specifically stimulated emission in our engineered E. coli cells. We will do this by determining the minimal concentration of fluorophores to reach the lasing threshold, when assuming the cavity has a fixed size (size of E. coli). We will not only determine when the lasing threshold is reached at a fixed size of the cell, but we will also determine the cavity size at which the lasing threshold is reached for a fixed concentration of fluorophores. We will determine these threshold conditions only for wild type GFP since the required parameters of GFP are most easily found. In this model we are determining the order of magnitude of the lasing threshold. Since all fluorophores we use are derived from GFP we assume that they will reach lasing in the same order of magnitude.

Fluorophores between high reflective mirrors

If the concentration of fluorophores is not large enough there will not be a burst of photons, as most of the photons are lost by escape through the mirrors or absorption in the medium and do not collide with excited molecules often enough. When only spontaneous emission takes place, \(GFP_1\) relaxes with a relaxation time \(\tau_{sp}\) (figure 4A). When there is stimulated emission there are enough photons within the system to trigger a burst of photons in a short time, much shorter than the spontaneous relaxation time (figure 4B).

Figure 4: The dynamics of excited fluorophores and number of photons at emission wavelength over time for fluorophores placed between two mirrors at a distance of 0.7 mm. All fluorophores are put in the excited state at t=0. (A) Relaxation of fluorophores through spontaneous emission of photons at a fluorophore concentration of \(1\mu M\). The blue line is the number of excited fluorophores within the system, which decreases exponentially over time. The red line is the number of photons at emission wavelength within the system. (B) Relaxation of fluorophores through stimulated emission of photons at a fluorophore concentration of \( 25 \mu M\). The number of excited fluorophores in the system (blue circles) drops dramatically within a very short time window of 0.1 ns. The number of photons (red circles) increase initially until most of the fluorophores went to the ground state, from that point there are no new photons produced and the number of photons leaving the system drops exponentially.

We calculated the photon dynamics for GFP molecules in a system of two mirrors at reflectivity R=0.9, with a spacing of d=0.7mm. When plotting the maximal output intensity versus concentration on a log-log scale (figure 5), we can see that the intensity suddenly increases around \(C=10\mu M\). We expect that this increase is due to the system transitioning from mainly spontaneous emission to mainly stimulated emission. This threshold concentration is lower than concentrations where lasing has been observed in a similar setup (Gather et al., 2011), showing that our model could be a good approach of predicting the order of magnitude where lasing takes place. Before and after the threshold is reached the output intensity is linear with the concentration of fluorophores.

Figure 5: Output energy as a function of concentration of fluorophores.

Threshold concentration in a single cell laser

To calculate the photon dynamics in our own system, where we use a bacterium as an optical cavity and a layer of polysilicate/tin dioxide as reflective agent, we use the same model as described above, given by equations 10 and 11.

The cavity size in this system is much smaller than in the case discussed above where distance between the mirrors was 0.7 mm, while the size of a bacterium is about 1 µm in diameter. Therefore the number of times a photon hits a mirror per distance traveled increases significantly. Furthermore we do not use mirrors with high reflectivity to trap the photons but a layer of polysilicate or tin dioxide. We take the reflectivity of a thin layer of these materials into account as described above (eqs. 12-21, table 1). There is a larger chance of escape with every incidence with the polysilicate or tin dioxide surface, compared to a high reflective mirror. Therefore the mirror losses in the system, consisting of a single bacteria, are larger than the mirror losses by a high reflective mirror. All in all, in this system the losses are larger than the system described above and therefore we expect that the laser threshold will be at a higher concentration of fluorophores.

For a system size of 1 μm and a reflectivity as given in table 1, our model shows that the lasing threshold has increased significantly (Figure 6) compared to the laser consisting of 2 high reflective mirrors spaced with a distance of 7 mm. We find the lasing threshold to be around 0.1 M for both materials of the shell. Furthermore we can see that the threshold is reached in a smaller regime of the concentration.

Figure 6: Output energy as a function of concentration of fluorophores in cells encapsulated by (a) polysilicate and (b) Tin dioxide

Threshold cell size in a single cell laser

By comparing the experimental data with a calibration curve in Q2 , we find a maximal concentration of 20 mM of fluorophores inside our cells. This is however too low to reach the lasing threshold for a laser cavity with a diameter of 1 µm. Therefore we determine at what cell size the lasing threshold can be reached at a fluorophore concentration of 20 mM (figure 7). In figure 7 we can clearly see that the lasing threshold is reached at a cell size around 8 µm, which is comparable to the smallest length scale used for organic lasers (Humar et al., 2015).

Figure 7: Output energy as a function of threshold cavity size for single cell lasers, showing a threshold value at about 8 µm.

Conclusion

In this model we determine at which concentration and which cell size the threshold is reached. We found that the minimal concentration in a cell of 1 µm in diameter is about 0.1 M. This concentration is a factor 10 larger than concentrations found in the cell. The minimal cavity size found to get lasing at a concentration of 20 mM (concentration we found in our cells) is about 8 µm. To acquire a single cell laser, we require using a different type of cell which has a larger size, such as mammal cells.

Download

The files used in this model can be downloaded here

References

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